Dimension of overlapping self-similar and self-conformal sets

One of the main problems in fractal geometry is computing the dimension of self-similar and self-conformal (cookie-cutters) when there are complicated overlaps. The exact overlaps conjecture predicts that, at least in the analytic case, the dimension is the "expected one" (the minimum between 1 and the zero of the Bowen pressure) unless the semigroup generated by the maps in the IFS is not free (there is an exact overlap). I will survey some dramatic progress towards this conjecture in the linear and non-linear setting by Hochman, Varju and Rapaport, among others.